Several techniques are used to model multidimensional data by mapping multidimensional input values to multidimensional output values. Such models often are used to recognize hidden predictive patterns in a data set. The kinds of problems for which a model may be used include clustering, classification and estimation of data in the data set. There are several types of models that are commonly used, such as probabilistic neural networks, generalized regression neural networks, Gaussian radial basis functions, decision trees (such as, K-D trees, neural trees and classification and regression trees), neural networks, Kohonen networks and associative algorithms.
Most modeling techniques are procedural but not declarative. In other words, a model maps input values to output values. This mapping does not convey the actual meaning or significance of what the model is doing, i.e., its behavior. It is difficult to predict how the model behaves in response to new inputs or what dimensions of the input are most relevant to the behavior of the model.
This problem is compounded when the input data includes a large number of dimensions. In order to ensure that a model is based on relevant input dimensions, various statistical techniques are used to analyze a data set that will be used to create a model in order to identify those dimensions that are salient to the problem to be modeled. A model is created using only the salient dimensions for the input. Example statistical techniques for identifying these salient dimensions include chi-squared automatic interaction detection (CHAID), correlation, principle component analysis, and sensitivity analysis.
Such techniques for identifying the salient dimensions used to create a model still do not provide an explanation of the behavior of the created model. In particular, some dimensions may be salient only in a subspace of the input data and therefore have an impact on the behavior of the model only in that subspace. To assist in understanding the behavior of a model, another kind of statistical technique, called rule induction, often is used. Rule induction is described, for example, in C4.5: Programs for Machine Learning, by J. Ross Quinlan, Morgan Kaufman Publishers, 1993. A computer program having the same name ("C4.5") also is available from that author and publisher. This program uses data directly to derive rules. Other rule induction techniques use a model to derive rules. These techniques provide a tree structure that explains the behavior of a model as a collection of rules. Although these rules may help to explain the behavior of the model, the rules often are too numerous and too complex for a human to interpret as easily as one would like. It also is difficult to extract from these rules an explanation of which input values are important in each subspace of the input data that the tree defines.